Step |
Hyp |
Ref |
Expression |
1 |
|
qtopcmp.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
qtopcmplem.1 |
⊢ ( 𝐽 ∈ 𝐴 → 𝐽 ∈ Top ) |
3 |
|
qtopcmplem.2 |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 ) |
4 |
|
simpl |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐽 ∈ 𝐴 ) |
5 |
|
simpr |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 Fn 𝑋 ) |
6 |
|
dffn4 |
⊢ ( 𝐹 Fn 𝑋 ↔ 𝐹 : 𝑋 –onto→ ran 𝐹 ) |
7 |
5 6
|
sylib |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 : 𝑋 –onto→ ran 𝐹 ) |
8 |
1
|
qtopuni |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ ran 𝐹 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) ) |
9 |
2 8
|
sylan |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 : 𝑋 –onto→ ran 𝐹 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) ) |
10 |
6 9
|
sylan2b |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) ) |
11 |
|
foeq3 |
⊢ ( ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) → ( 𝐹 : 𝑋 –onto→ ran 𝐹 ↔ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ( 𝐹 : 𝑋 –onto→ ran 𝐹 ↔ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ) ) |
13 |
7 12
|
mpbid |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ) |
14 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
15 |
2 14
|
sylib |
⊢ ( 𝐽 ∈ 𝐴 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
16 |
|
qtopid |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) |
17 |
15 16
|
sylan |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) |
18 |
4 13 17 3
|
syl3anc |
⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 ) |